Certainly! Below is the transcription suitable for an educational website:---**Consider the following.****Differential Equation**\[ y''' + 6y'' + 9y' = 0 \]**Solutions**\[ \{e^{-3x}, xe^{-3x}, (3x + 1)e^{-3x}\} \]**(a) Verify that each solution satisfies the differential equation.**1. \( y = e^{-3x} \) - \( y' = \) [Box for entering derivative] - \( y'' = \) [Box for entering second derivative] - \( y''' = \) [Box for entering third derivative] - \( y''' + 6y'' + 9y' = \) [Box for showing verification]2. \( y = xe^{-3x} \) - \( y' = \) [Box for entering derivative] - \( y'' = \) [Box for entering second derivative] - \( y''' = \) [Box for entering third derivative] - \( y''' + 6y'' + 9y' = \) [Box for showing verification]3. \( y = (3x + 1)e^{-3x} \) - \( y' = \) [Box for entering derivative] - \( y'' = \) [Box for entering second derivative] - \( y''' = \) [Box for entering third derivative] - \( y''' + 6y'' + 9y' = \) [Box for showing verification]--- This exercise involves verifying the given solutions by computing their derivatives and substituting them back into the original differential equation to ensure they satisfy it. The image displays an exercise related to differential equations. The task involves solving the following problems:1. **Differential Equation:** - \( y''' + 6y'' + 9y' = \) [Input Box]2. **Linear Independence Test:** - The user must test the set of solutions for linear independence by choosing one of the two options: - ( ) linearly independent - ( ) linearly dependent3. **General Solution:** - If the set is deemed linearly independent, users are to input the general solution of the differential equation. Instructions note that if the system is dependent, the user should enter "DEPENDENT." Constants \( C_1 \) and \( C_2 \) may be used as needed in the general solution: - \( y = \) [Input Box]4. **Submission Button:** - "Submit Answer" is mentioned at the bottom for submitting responses.There are no graphs or diagrams present in the image.

Answered: Consider the following. Differential…

Consider the following. Differential Equation Solutions y"" + 6y" +9y' = 0 {e-3x, xe-3x, (3x + 1)e-³x) (a) Verify that each solution satisfies the differential equation. y = e-3x

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

Solve the differential equation for general solution y' = 2x - xy; y = 5 when x = O y = -3ex²/2+ 8 O y = 5ex?/2 O y = 7ex/2 + 5 O y = 3ex²/2 + 2
Solve for Differential Equations:
Form a differential equation by eliminating the arbitrary constant a and b from the equation y = ax² 2bx
b) Solve differential equation x y³ý = x* + 2 y*